If you can, freeze the base and add the yogurt later. If it is meant to be served cold, let it defrost slowly (in the fridge would be good) when you're going to serve it. If the yogurt is already mixed in, try it -- what do you have to lose? I would whisk it just before putting in bowls.

Of course you can freeze yogurt! I often put those yogurt tubes the kids like in the freezer for snacks and if cups are on sale I but extra and freeze. There should be no issue but if does separate slightly just stir again to combine.